Abstract

Let $K$ be a quadratic number field with ring of integers $\mathcal{O}_K$. It is well known that the class number $h_K$ is one if and only if $\mathcal{O}_K$ is a principal ideal domain. While the imaginary quadratic fields of class number one have been completely classified, Gauss’s class number one conjecture, which predicts the existence of infinitely many real quadratic fields of class number one, remains one of the central open problems in algebraic number theory.

In this lecture, we will discuss the divisibility of class numbers of quadratic number fields by a prescribed integer. A classical result, established through the work of several authors, states that for every positive integer $n$, there exist infinitely many quadratic fields $K$ whose class numbers are divisible by $n$. Motivated by this phenomenon, Iizuka conjectured that for every integer $m \geq 1$ and every prime $\ell \geq 3$, there exist infinitely many tuples $$\mathbb{Q}(\sqrt{d}),\ \mathbb{Q}(\sqrt{d+1}),\ \ldots,\ \mathbb{Q}(\sqrt{d+m}),$$ of real (or imaginary) quadratic fields, where $d \in \mathbb{Z}$, such that $\ell$ divides the class number of each field in the tuple.

In this lecture, I will present a partial result toward this conjecture by establishing the existence of infinitely many triples of imaginary quadratic fields whose class numbers are all divisible by $3$. I will conclude by discussing the main ideas behind the proof and highlighting several intriguing open problems and possible directions for future research.

This is joint work with Jaitra Chattopadhyay.